Generalized exterior sphere conditions and \varphi-convexity

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April 2011, 29(2): 615-622. doi: 10.3934/dcds.2011.29.615

Generalized exterior sphere conditions and $\varphi$-convexity

Chadi Nour ^1, , Ron J. Stern ^2, and Jean Takche ^3,

1.	Computer Science and Mathematics Division, Lebanese American University, Byblos Campus, P.O. Box 36, Byblos
2.	Department of Mathematics and Statistics, Concordia University, 1400 De Maisonneuve Blvd. West, Montreal, Quebec H3G 1M8, Canada
3.	Department of Computer Science and Mathematics, Lebanese American University, Byblos Campus, P.O. Box 36, Byblos, Lebanon

Received August 2009 Revised March 2010 Published October 2010

Abstract
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We consider sets $S\subset\R^n$ satisfying a certain exterior sphere condition, and it is shown that under wedgedness of $S$, it coincides with $\varphi$-convexity. We also offer related improvements concerning the union of uniform closed balls conjecture.

Keywords: $\psi$-semiconcavity, $\theta$-exterior sphere condition, proximal analysis, nonsmooth analysis., wedgedness, union of uniform closed balls conjecture, $\varphi$-convexity.

Mathematics Subject Classification: Primary: 49J52; Secondary: 52A2.

Citation: Chadi Nour, Ron J. Stern, Jean Takche. Generalized exterior sphere conditions and $\varphi$-convexity. Discrete and Continuous Dynamical Systems, 2011, 29 (2) : 615-622. doi: 10.3934/dcds.2011.29.615

References:

[1]	P. Cannarsa and H. Frankowska, Interior sphere property of attainable sets and time optimal control problems, ESAIM: Control Optim. Calc. Var., 12 (2006), 350-370. doi: doi:10.1051/cocv:2006002.
[2]	P. Cannarsa and C. Sinestrari, Convexity properties of the minimum time function, Calc. Var., 3 (1995), 273-298. doi: doi:10.1007/BF01189393.
[3]	P. Cannarsa and C. Sinestrari, "Semiconcave Functions, Hamiton-Jacobi Equations and Optimal Control," Birkhäser, Boston, 2004.
[4]	A. Canino, On $p$-convex sets and geodesics, J. Differential Equations, 75 (1988), 118-157. doi: doi:10.1016/0022-0396(88)90132-5.
[5]	F. H. Clarke, "Optimization and Nonsmooth Analysis," Classics in Applied Mathematics, 5, SIAM, Philadelphia, PA, 1990.
[6]	F. H. Clarke, Yu. Ledyaev, R. J. Stern and P. R. Wolenski, "Nonsmooth Analysis and Control Theory," Graduate Texts in Mathematics, 178, Springer-Verlag, New York, 1998.
[7]	F. H. Clarke and R. J. Stern, State constrained feedback stabilization, SIAM J. Control Optim., 42 (2003), 422-441. doi: doi:10.1137/S036301290240453X.
[8]	F. H. Clarke, R. J. Stern and P. R. Wolenski, Proximal smoothness and the lower-$C^2$ property, J. Convex Anal., 2 (1995), 117-144.
[9]	G. Colombo and A. Marigonda, Differentiability properties for a class of non-convex functions, Calc. Var., 25 (2005), 1-31. doi: doi:10.1007/s00526-005-0352-7.
[10]	G. Colombo, A. Marigonda and P. R. Wolenski, Some new regularity properties for the minimal time function, SIAM J. Control Optim., 44 (2006), 2285-2299. doi: doi:10.1137/050630076.
[11]	H. Federer, Curvature measures, Trans. Amer. Math. Soc., 93 (1959), 418-491.
[12]	C. Nour, R. J. Stern and J. Takche, Proximal smoothness and the exterior sphere condition, J. Convex Anal., 16 (2009), 501-514.
[13]	C. Nour, R. J. Stern and J. Takche, The $\theta$-exterior sphere condition, $\varphi$-convexity, and local semiconcavity, Nonlinear Anal. Theor. Meth. Appl., 73 (2010), 573-589. doi: doi:10.1016/j.na.2010.04.001.
[14]	R. A. Poliquin and R. T. Rockafellar, Prox-regular functions in variational analysis, Trans. Amer. Math. Soc., 348 (1996), 1805-1838. doi: doi:10.1090/S0002-9947-96-01544-9.
[15]	R. A. Poliquin, R. T. Rockafellar and L. Thibault, Local differentiability of distance functions, Trans. Amer. Math. Soc., 352 (2000), 5231-5249. doi: doi:10.1090/S0002-9947-00-02550-2.
[16]	R. T. Rockafellar, Clarke's tangent cones and the boundaries of closed sets in $\R^n$, Nonlinear Anal. Theor. Meth. Appl., 3 (1979), 145-154. doi: doi:10.1016/0362-546X(79)90044-0.
[17]	R. T. Rockafellar and R. J.-B. Wets, "Variational Analysis," Grundlehren der Mathematischen Wissenschaften, 317, Springer-Verlag, Berlin, 1998.
[18]	A. S. Shapiro, Existence and differentiability of metric projections in Hilbert spaces, SIAM J. Optim., 4 (1994), 231-259. doi: doi:10.1137/0804006.
[19]	C. Sinestrari, Semiconcavity of the value function for exit time problems with nonsmooth target, Commun. Pure Appl. Anal., 3 (2004), 757-774. doi: doi:10.3934/cpaa.2004.3.757.

show all references

References:

[1]	P. Cannarsa and H. Frankowska, Interior sphere property of attainable sets and time optimal control problems, ESAIM: Control Optim. Calc. Var., 12 (2006), 350-370. doi: doi:10.1051/cocv:2006002.
[2]	P. Cannarsa and C. Sinestrari, Convexity properties of the minimum time function, Calc. Var., 3 (1995), 273-298. doi: doi:10.1007/BF01189393.
[3]	P. Cannarsa and C. Sinestrari, "Semiconcave Functions, Hamiton-Jacobi Equations and Optimal Control," Birkhäser, Boston, 2004.
[4]	A. Canino, On $p$-convex sets and geodesics, J. Differential Equations, 75 (1988), 118-157. doi: doi:10.1016/0022-0396(88)90132-5.
[5]	F. H. Clarke, "Optimization and Nonsmooth Analysis," Classics in Applied Mathematics, 5, SIAM, Philadelphia, PA, 1990.
[6]	F. H. Clarke, Yu. Ledyaev, R. J. Stern and P. R. Wolenski, "Nonsmooth Analysis and Control Theory," Graduate Texts in Mathematics, 178, Springer-Verlag, New York, 1998.
[7]	F. H. Clarke and R. J. Stern, State constrained feedback stabilization, SIAM J. Control Optim., 42 (2003), 422-441. doi: doi:10.1137/S036301290240453X.
[8]	F. H. Clarke, R. J. Stern and P. R. Wolenski, Proximal smoothness and the lower-$C^2$ property, J. Convex Anal., 2 (1995), 117-144.
[9]	G. Colombo and A. Marigonda, Differentiability properties for a class of non-convex functions, Calc. Var., 25 (2005), 1-31. doi: doi:10.1007/s00526-005-0352-7.
[10]	G. Colombo, A. Marigonda and P. R. Wolenski, Some new regularity properties for the minimal time function, SIAM J. Control Optim., 44 (2006), 2285-2299. doi: doi:10.1137/050630076.
[11]	H. Federer, Curvature measures, Trans. Amer. Math. Soc., 93 (1959), 418-491.
[12]	C. Nour, R. J. Stern and J. Takche, Proximal smoothness and the exterior sphere condition, J. Convex Anal., 16 (2009), 501-514.
[13]	C. Nour, R. J. Stern and J. Takche, The $\theta$-exterior sphere condition, $\varphi$-convexity, and local semiconcavity, Nonlinear Anal. Theor. Meth. Appl., 73 (2010), 573-589. doi: doi:10.1016/j.na.2010.04.001.
[14]	R. A. Poliquin and R. T. Rockafellar, Prox-regular functions in variational analysis, Trans. Amer. Math. Soc., 348 (1996), 1805-1838. doi: doi:10.1090/S0002-9947-96-01544-9.
[15]	R. A. Poliquin, R. T. Rockafellar and L. Thibault, Local differentiability of distance functions, Trans. Amer. Math. Soc., 352 (2000), 5231-5249. doi: doi:10.1090/S0002-9947-00-02550-2.
[16]	R. T. Rockafellar, Clarke's tangent cones and the boundaries of closed sets in $\R^n$, Nonlinear Anal. Theor. Meth. Appl., 3 (1979), 145-154. doi: doi:10.1016/0362-546X(79)90044-0.
[17]	R. T. Rockafellar and R. J.-B. Wets, "Variational Analysis," Grundlehren der Mathematischen Wissenschaften, 317, Springer-Verlag, Berlin, 1998.
[18]	A. S. Shapiro, Existence and differentiability of metric projections in Hilbert spaces, SIAM J. Optim., 4 (1994), 231-259. doi: doi:10.1137/0804006.
[19]	C. Sinestrari, Semiconcavity of the value function for exit time problems with nonsmooth target, Commun. Pure Appl. Anal., 3 (2004), 757-774. doi: doi:10.3934/cpaa.2004.3.757.

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